       Variable section and hollow Tubes for economic 3D printing

• To: mathgroup at smc.vnet.net
• Subject: [mg132557] Variable section and hollow Tubes for economic 3D printing
• From: Narasimham <mathma18 at gmail.com>
• Date: Fri, 11 Apr 2014 02:10:49 -0400 (EDT)
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```TubeRadius = 0.2 ; X[u_] = {Cos[u] , Sin[u], 0 };
CentrCirc =
ParametricPlot3D[X[u], {u, 0, 2 Pi}, PlotStyle -> {Red, Thick}];
Solid = ParametricPlot3D[ X[t], {t, 0,  Pi},PlotStyle -> Tube[TubeRadius], PlotRange -> All];
Show[{CentrCirc, Solid}, PlotRange -> All]
" The above is a solid torus by inclusive consideration of all TR values between 0 to 0.2, or, so to say < solidifies > the Surface given below that is valid for outer radius TR = 0.2 only. "
Surface = ParametricPlot3D[ {(a + TubeRadius Cos[ph]) Cos[t], (a + TubeRadius Cos[ph]) Sin[t],TubeRadius Sin[ph]}, {ph, -Pi, Pi}, {t, 0,  Pi}];
Show[{Surface, CentrCirc}, PlotRange -> All]
" However, if TubeRadius is made function of t or ph how to use Tube[TubeRadius], between limits TR = 0 and TR = 0.2, or even between limits TR = 0.1 and TR = 0.2 hollow, in order to <solidify> the surface ? "
TR[u_] = 0.2 + 0.15 Cos[u] ;
a = 1; Plot[TR[t], {t, 0, 2 Pi}]
ParametricPlot3D[ {(a + TR[t] Cos[ph]) Cos[t], (a + TR[t] Cos[ph]) Sin[t], 0 t + TR[t] Sin[ph]}, {ph, -Pi, Pi}, {t, 0,  Pi}]
ParametricPlot3D[ {(a + TR[t] Cos[ph]) Cos[t], (a + TR[t] Cos[ph]) Sin[t], .8 t + TR[t] Sin[ph]}, {ph, -Pi,Pi}, {t, 0,  Pi}]
ParametricPlot3D[ {(a + TR[2 ph] Cos[ph]) Cos[t], (a + TR[2 ph] Cos[ph]) Sin[t], .8 t + TR[2 ph] Sin[ph]}, {ph, -Pi, Pi}, {t, 0, Pi}]

After all, a solid is formed by surface growth along its normal. It is believed such a generalization command in Mathematica for all surfaces would be of help in 3D printing of plastic mathematical models that are presently fast catching on.

Best Regards
Narasimham GL "

```

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